The general Characteristics of the frequencyfunction of stellar movements 39 



Accordingly the correlation tables of two such diametrically situated squares 

 may be added (when changing the sign of the proper motion in right ascension on 

 the southern hemisphere) and the moments computed for the motions. The mo- 

 ments given in table I are the moments of the apparent motion about the mean 

 of such a double square. In the following I shall always refer to the double- 

 squares as shortly squares, thus by C l meaning the double-cone having the solid 

 angle C v Similarly the other double-squares are denoted by the name of its nor- 

 thern component. 



Now, one objection may be raised against this arrangement. When, namely, 

 the system of stars as a whole should be affected by a rotational motion about 

 some axis, then clearly the » spread » of the motions in our compound squares 

 should be greatened, as the rotation must affect the two components in opposite 

 directions. Consequently we should find our second and fourth moments too great. 

 Indeed, Charlier, has found, for the stars here employed, such a rotational motion 

 amounting to p = 0".0035 per year, about an axis nearly normal to the plane 

 of the Milky Way*. This rotational motion is, however, small enough to be 

 altogether neglected when taking together the stars in diametrical squares. Assuming, 

 for instance, the number of stars in the two component squares to be equal, we 

 should have to correct v 20 by ■ — p 2 sin 2 <p; v 02 by — p 8 cos 2 ip; v u by — p 2 sin <p cos <p. 

 The moments of the third order are not affected at all and by those of the fourth 

 order we should have to correct, for instance, v 40 — 3v. J0 2 by -f-p 4 sin 4 f. For <p we 

 have to take the angle between the ^r-axis of the square and the axis of rotation' 

 and to find the above corrections formulae (68*) (68**) and (68***) can be used. 

 As p = — i we see that taking the classbreadth as unity the corrections of the second 

 and fourth characteristics at most amount to 0.0051 and 0.000026, which can be wholly 

 neglected, especially as we have already neglected the corrections of Sheppard for 

 the classbreadth, which affect even one higher of the decimal places. 



The direction cosines of the centres of gravity of the squares are according to 

 Charlier** as put together in table V. 



* This is the motion of the node of the invariable plane of the solar system upon the 

 plane of the Galaxy. It is direct and has a period of about 370 million years. 



** The direction cosines for the TJ- and 7-axes tabulated by Charlier 1. c. p. 70 we must 

 take with changement of sign. 



